# Robust formula for $N$-point Pad\'e approximant calculation based on Wynn identity

## TL;DR

This paper introduces a new criterion based on Wynn's identity for selecting the optimal N-point Padé approximant, enhancing the accuracy and reliability of rational approximations in physics and applied mathematics.

## Contribution

The work presents a novel formula and criterion for optimal Padé approximation using Wynn's identity, with practical applications demonstrated in physics and differential equations.

## Key findings

- Wynn's identity provides a criterion for minimal |η| as the optimal Padé approximant
- The method improves multipoint Padé approximation accuracy
- Application to physics problems like solar corona heating series summation

## Abstract

The performed numerical analysis reveals that Wynn's identity for the compass $1/(N-C)+1/(S-C)=1/(W-C)+1/(E-C)=1/\eta$ (here C stands for center, the other letters correspond to the four directions of the compass) gives the long sought criterion, the minimal $|\eta|$, for the choice of the optimal Pad\'e approximant. The work of this method is illustrated by calculation of multipoint Pad\'e approximation by a new formula for calculation of this best rational approximation. The work of the criterion for the calculation of optimal Pad\'e approximant is illustrated by the frequently seen in the theoretical physics problems of calculation of series summation and multipoint Pad\'e approximation used as a predictor for solution of differential equations motivated by the magneto-hydrodynamic problem of heating of solar corona by Alv\'en waves. In such a way, an efficient and valuable control mechanism for $N$-point Pad\'e approximant calculation is proposed. We believe that the suggested method and criterion can be useful for many applied problems in numerous areas not only in physics but in any scientific application where differential equations are solved. The solution of the Cauchy-Jacobi problem is illustrated by a Fortran program. The algorithm is generalized for the case of the first $K$ derivatives at $N$ nodal points.